Problem: A group of adults and kids went to see a movie. Tickets cost $$8.00$ each for adults and $$4.50$ each for kids, and the group paid $$72.50$ in total. There were $5$ fewer adults than kids in the group. Find the number of adults and kids in the group.
Let $x$ equal the number of adults and $y$ equal the number of kids. The system of equations is then: ${8x+4.5y = 72.5}$ ${x = y-5}$ Solve for $x$ and $y$ using substitution. Since $x$ has already been solved for, substitute ${y-5}$ for $x$ in the first equation. ${8}{(y-5)}{+ 4.5y = 72.5}$ Simplify and solve for $y$ $ 8y-40 + 4.5y = 72.5 $ $ 12.5y-40 = 72.5 $ $ 12.5y = 112.5 $ $ y = \dfrac{112.5}{12.5} $ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into ${x = y-5}$ to find $x$ ${x = }{(9)}{ - 5}$ ${x = 4}$ You can also plug ${y = 9}$ into ${8x+4.5y = 72.5}$ and get the same answer for $x$ ${8x + 4.5}{(9)}{= 72.5}$ ${x = 4}$ There were $4$ adults and $9$ kids.